Answered

Simplify the radical: $\sqrt{80 x^{10} y^4}$


A$4 x y \sqrt{5}$
B$4 x^2 y^2 \sqrt{5}$
C$4 x^b y^2 \sqrt{5 x y}$
D$4 x^2 y^2 \sqrt{5 x}$

Verified Answer :

  • Factor 80 into the product of a perfect square and a square-free integer: \(80 = 16 \times 5\).
  • Express the variables as powers of squares: \(x^{10} = (x^5)^2\) and \(y^4 = (y^2)^2\).
  • Apply the square root property \(\sqrt{abc} = \sqrt{a}\sqrt{b}\sqrt{c}\) to simplify each part.
  • The result is \(4x^5y^2\sqrt{5}\), which corresponds to choice \(\boxed{\text{(C)}}\).

Explanation

  1. Analyzing the Radicand

To simplify the expression \(\sqrt{80x^{10}y^4}\), we begin by breaking down the expression inside the radical (the radicand) into its prime factors and identifying any perfect square components. The radicand is \(80x^{10}y^4\).
We can factor 80 as:
\[80 = 16 \times 5 = 4^2 \times 5\]
Next, consider the variables with exponents. We want to identify the largest even powers within the exponents, as taking the square root of an even power \(x^{2n}\) gives \(x^n\):

  • For \(x^{10}\), it is already a perfect square since \(10\) is even: \(x^{10} = (x^5)^2\).
  • For \(y^4\), it is also a perfect square since \(4\) is even: \(y^4 = (y^2)^2\).
  1. Applying Square Root Properties

Now, we substitute these components back into the radical expression and apply the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) for non-negative \(a\) and \(b\):
\[\sqrt{80x^{10}y^4} = \sqrt{16 \cdot 5 \cdot x^{10} \cdot y^4}\]
\[\sqrt{80x^{10}y^4} = \sqrt{16} \cdot \sqrt{x^{10}} \cdot \sqrt{y^4} \cdot \sqrt{5}\]

  1. Simplifying the Expression and Selecting the Correct Option

Calculate the square root of each perfect square term:

  • \(\sqrt{16} = 4\)
  • \(\sqrt{x^{10}} = x^{10/2} = x^5\)
  • \(\sqrt{y^4} = y^{4/2} = y^2\)

Combining these results, we get:
\[4 \cdot x^5 \cdot y^2 \cdot \sqrt{5}\]
This simplifies to:
\[4x^5y^2\sqrt{5}\]
Comparing this result with the given options, it matches choice (C).

  1. Conclusion

The final simplified form of the expression \(\sqrt{80x^{10}y^4}\) is \(4x^5y^2\sqrt{5}\).
Therefore, the correct choice is \(\boxed{\text{(C)}}\).

Examples

In fields like physics or engineering, simplifying radical expressions is essential when working with formulas that involve distances, magnitudes of vectors, or time dilation calculations. For instance, if you are calculating the displacement or the velocity of an object using Pythagorean-based relationships, you often end up with complex radicals. Simplifying these expressions allows you to compare results more easily and identify relationships between different physical quantities, similar to how we simplified the expression in the problem to find the most compact representation.

Related Questions